A Question of Erd\H{o}s and Graham on Covering Systems

TL;DR

This paper resolves an Erd ext{"o}s-Graham question by proving no such integer n exists where divisors greater than 1 form a covering system with pairwise coprime moduli, and explores related divisor systems.

Contribution

It proves the non-existence of an integer n with the specified covering system property and analyzes conditions under which divisors of n can form such systems.

Findings

No such n exists for the covering system with coprime moduli.

Conditions for divisors of n to form such systems are characterized.

The problem is connected to a known Erd ext{"o}s-Graham question.

Abstract

Erd\H{o}s and Graham (Erd\H{o}s and Graham, 1980) asked if there exists an $n$ such that the divisors of $n$ greater than 1 are the moduli of a distinct covering system with the following property: If there exists an integer which satisfies two congruences in the system, $a\mod d$ and $a'\mod d'$, then $\gcd(d,d')=1$. We show that such an $n$ does not exist. This problem is part of Problem # 204 on the website www.erdosproblems.com, compiled and maintained by Thomas Bloom. We also study when the divisors of $n$ greater than $1$ can form a congruence system satisfying the above condition.